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Transcript of Mathematics Stage 5 MS5.1.2...
Mathematics Stage 5
MS5.1.2 Trigonometry
Part 3 Applying trigonometry
Number: M43683 Title: MS5.1.2 Trigonometry
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Published byCentre for Learning Innovation (CLI)51 Wentworth RdStrathfield NSW 2135________________________________________________________________________________________________Copyright of this material is reserved to the Crown in the right of the State of New South Wales. Reproduction ortransmittal in whole, or in part, other than in accordance with provisions of the Copyright Act, is prohibited withoutthe written authority of the Centre for Learning Innovation (CLI).
© State of New South Wales, Department of Education and Training 2005.
This publication is copyright New South Wales Department of Education and Training (DET), however it may containmaterial from other sources which is not owned by DET. We would like to acknowledge the following people andorganisations whose material has been used:
Extracts from Mathematics Syllabus Years 7-10 ©Board of Studies, NSW 2002 Unit overview pp iii-iv, Part 1 p 3, Part 2p 3, Part 3 p 3-4
COMMONWEALTH OF AUSTRALIA
Copyright Regulations 1969
WARNING
This material has been reproduced and communicated to you on behalf of theNew South Wales Department of Education and Training
(Centre for Learning Innovation)pursuant to Part VB of the Copyright Act 1968 (the Act).
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CLI Project Team acknowledgement:
Writer(s): James StamellEditor(s): Ric MoranteIllustrator(s): Thomas Brown, Tim HutchinsonDesktop Publisher(s): Gayle ReddyVersion date: April 27, 2005
Part 3 Applying trigonometry 1
Contents – Part 3
Introduction – Part 3..........................................................3
Indicators ...................................................................................3
Preliminary quiz.................................................................5
Choosing the ratio .............................................................9
General problems............................................................13
Gradient of a line.....................................................................16
Elevation and depression................................................19
Further elevation and depression....................................23
Reviewing trigonometry...................................................27
Suggested answers – Part 3 ...........................................29
Exercises – Part 3 ...........................................................33
2 MS5.1.2 Trigonometry
Part 3 Applying trigonometry 3
Introduction – Part 3
This third part of trigonometry deals with deciding which of the three
trigonometric functions to use in answering a question, and to solve
problems where trigonometry can be applied. It is also extended to cover
angles of elevation and depression.
Throughout these notes diagrams are given. Occasionally incomplete
diagrams are provided to encourage students to add relevant information
to them which will aid them in answering the question.
Indicators
By the end of Part 3, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
• selecting and using appropriate trigonometric ratios in right-angled
triangles to find unknown sides, including the hypotenuse
• selecting and using appropriate trigonometric ratios on right-angled
triangles to find unknown angles correct to the nearest degree
• identifying angles of elevation and depression
• solving problems involving angles of elevation and depression when
given a diagram.
By the end of Part 3, you will have been given the opportunity to work
mathematically by:
• solving problems in practical situations involving right-angled
triangles
• interpreting diagrams in questions involving angles of elevation and
depression
4 MS5.1.2 Trigonometry
• relating the tangent ratio to the gradient of a line.Source: Extracts from outcomes of the Mathematics Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pdf > (accessed 04 November 2003).© Board of Studies NSW, 2002.
Part 3 Applying trigonometry 5
Preliminary quiz
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1 Use your calculator to find, correct to 3 decimal places,
a sin 29° _____________________________________________
b tan 10° _____________________________________________
c cos 53° ____________________________________________
d4 × cos25°
5 _________________________________________
2 Find angles for which,
a sin A = 0.563 ________________________________________
b tan B = 12 __________________________________________
c cos C = 3
20 _________________________________________
d tan α = 1 ___________________________________________
6 MS5.1.2 Trigonometry
3 Use the cosine ratio to calculate the length of the side marked.
E F
G
x
126 m
36°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
4 Use the tangent ratio to calculate the length of the side marked.
e 7.6 cm
38°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
5 Use the sine ratio to calculate the size of the angle between north and
the diagonal line.
5
6
N
E
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Part 3 Applying trigonometry 7
6 Use the tangent ratio to calculate the size of the angle marked.
G H15.3 cm
9.5 cm
h°
F
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
8 MS5.1.2 Trigonometry
Part 3 Applying trigonometry 9
Choosing the ratio
Until now you have been directed to the trig function (sine, cosine, or
tangent) you needed to use. You have now reached a stage where you
should be able to decide which of these three trig ratios is the appropriate
one to use in any given situation.
SOH CAH TOA
sin opphyp cos adj
hyp tan oppadj op
pos
ite
adjacent
hypotenuse
*
For example, if the opposite side andhypotenuse are being used, the trig.ratio you want is sine.
Follow through the steps in this example. Do your own working in the
margin if you wish.
a Calculate the angle, a° , correct to the nearest degree.
B C
A
12 cm8 cm
a°
10 MS5.1.2 Trigonometry
b Calculate z correct to one decimal place.
Z Y
X17
cm z
25°
Solution
a The two sides involved are the adjacent (8 cm) and hypotenuse
(12 cm). Therefore use cosine.
cos a° = 8
12 ∴ a° = 48.189685°
= 48° (correct to the nearest degree)
use SHIFT cos
b This time the two sides involved are the opposite (17 cm) and the
hypotenuse (z). Sine is needed.
sin 25° = 17
z
z = 17
sin 25°= 81.8 cm (correct to one decimal place)
In each case, look over your answers to check if they are reasonable.
Activity – Choosing the ratio
Try these.
1 Use the appropriate trigonometric ratio to calculate x in each
triangle.
a
x
50°
80 mm
Part 3 Applying trigonometry 11
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b10
0 m
60°x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
2 Calculate the size of the marked angle, correct to the nearest degree.
a
38
α
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
41
9α
___________________________________________________
12 MS5.1.2 Trigonometry
___________________________________________________
___________________________________________________
___________________________________________________
c7 5
α β
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
You have been practising choosing and using the correct trig ratio in a
right-angled triangle. Now check that you can solve these kinds of
problems by yourself.
Go to the exercises section and complete Exercise 3.1 – Choosing the
ratio.
Part 3 Applying trigonometry 13
General problems
You have learned how to use the trigonometric ratios of an acute angle in
a right-angled triangle in two different ways.
• To find the size of the acute angles of the triangle
• To find the length of the side given one of the acute angles.
You will now use this knowledge to solve simple problems. In each case
you will need to use a labelled diagram to represent the situation. In this
set of notes a diagram will always be given, but you may want to add
more information to it.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Merill skis 132 metres down an even slope and drops 20 m in
height. What angle (to the nearest degree) does the slope make
with the horizontal?
132 m
α
20 m
14 MS5.1.2 Trigonometry
Solution
The diagram helps you identify that the sine ratio is required for
this question.
sin α = 20
132 ∴α = 8.715°
= 9° (to the nearest degree)
sin =opp.
hyp.
⎡
⎣⎢
⎤
⎦⎥
Try this calculation. Did you obtain the same answer?
Activity – General problems
Try these.
1 A ladder has its foot 1.3 metres from the base of a wall on level
ground. The ladder makes an angle of 75° with the horizontal.
Calculate the length of the ladder.
y
75°
1.3 m
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 3 Applying trigonometry 15
2 A boat is at anchor 500 m from the foot of a cliff 40 m high.
Calculate the angle, θ , which the line of sight to the top of the cliff
makes with the horizontal.
not drawn to scale
θ
40 m
500 m
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
In each of these examples you can see that, regardless of the situation, a
right-angled triangle is involved. Once you have identified that triangle
you then need to decide which of the three trig functions: sine cosine or
tangent you will use.
Use a diagram as a visual representation of the problem. It does not need
to be drawn to scale.
Check that the answer you obtain looks reasonable for the problem you
are solving. It is very easy to mistakenly choose the wrong trig ratio, or
press the wrong key on your calculator.
16 MS5.1.2 Trigonometry
Gradient of a line
In co-ordinate geometry you established that the slope of the line, m, is
given by m =rise
run.
Also from trigonometry,
tan θ = opposite
adjacent and so, from the
triangle shown, tan θ = rise
run.
This means that m = tan θ . O
run
rise
θ
y
x
The angle in the triangle is the same as the angle between the line and the
positive (right-hand) side of the x-axis. Can you see why? (Think of
parallel lines, and corresponding angles.)
So the gradient of a line is simply the tan of the angle it makes with the
positive side of the x-axis.
Follow through the steps in this example. Do your own working in the
margin if you wish.
What angle does the line y = 12
x + 1 make with the x-axis?
Solution
Make a small table of values to draw the line.
x –1 0 1 2
y 0.5 1 1.5 2
Now draw the line on the x-y plane and use it to determine the
gradient.
Part 3 Applying trigonometry 17
–1–2 1 2 30
–1
1
2
3 y
x
1
2θ
θ
The gradient, m = 1
2, and so tan θ =
1
2.
Using your calculator, θ = 27° (to the nearest degree).
You can use any two points on the line to establish rise and run.
Activity – General problems
Try this.
3 Draw the line y = 2x – 3 on the co-ordinate plane and use it to
determine the angle the line makes with the x-axis.
0
1
2
3
–1
–2
–3
–4
–5
–6
1 2 3–1
y
x
Check your response by going to the suggested answers section.
Go to the exercises section and complete Exercise 3.2 – General problems.
18 MS5.1.2 Trigonometry
Part 3 Applying trigonometry 19
Elevation and depression
If someone is looking at an object, the straight line from the eye of the
observer to the object is the line of sight.
line of sight
If the object is above the eye level of the observer, then the angle they
raise their eyes, from the horizontal to the line of sight of the object, is
called the angle of elevation.
horizontal at eye level
angle of elevation
line of sight
If the object is below the eye level of the observer, then the angle they
look down, from the horizontal to the line of sight of the object, is called
the angle of depression.
horizontal at eye level
angle of depression
line of sight
Note: the angle of elevation or depression of one point from another is
always measured from the horizontal.
20 MS5.1.2 Trigonometry
Since these horizontal lines are
parallel, the angle of elevation
must be equal to the angle of
depression, because they are
alternate angles formed by
parallel straight lines.
horizontal
horizontal
angle of depression
angle of elevation
You can use angle of elevation or angle of depression to answer
questions involving trigonometry.
Follow through the steps in this example. Do your own working in the
margin if you wish.
From a plane travelling at a height of 6 km, the angle of
depression of an airfield is found to be 14°. How far must the
plane fly to be directly above the airfield?
ground level
angle of depression14°
6 km
P Q
A(airfield)
(directly abovethe airfield)
Solution
∠PQA = 90° because the plane flies horizontally. At Q, it is
directly above A.
∆ PQA is the right-angled triangle. You want to calculate the
length PQ.
Part 3 Applying trigonometry 21
tan P = AQ
PQ
tan 14° = 6
PQ
PQ = 6
tan 14°= 24.065
(The two shorter sides involve using the tangent ratio.)
The plane must fly 24 km (to the nearest kilometre).
When answering questions like this, give your answer to an appropriate
number of significant figures. Correct to the nearest kilometre for this
question is reasonable, but writing 24.06469 km is not.
Activity – Elevation and depression
Try these.
1 From the top of a cliff 20 m high the angle of depression of a ship at
sea is 15°. Calculate the distance from the ship to the foot of the
cliff.
C
FS
15°75°
20 m
?15°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
22 MS5.1.2 Trigonometry
2 A flagpole of height 10 metres casts a shadow of length 16 metres on
level ground. Calculate the angle of elevation of the sun at this time.
10 m
16 m
F
EDx
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Notice that the angle of depression, in the first question, is the angle
outside the triangle. It is the angle measured downwards from the
horizontal.
You could use alternate angles, and work with ∠CSF inside the triangle.
Or you could calculate the complement of 15° (which is 75° ) and work
with ∠FCS. The choice is yours.
C
FS
15°75°
20 m
?15°
The two angles are equal.
The horizontal lines are parallel,and the angles are alternate. The two angles
add up to 90°.
Either way, the answer you will arrive at is the same.
Go to the exercises section and complete Exercise 3.3 – Elevation and
depression.
Part 3 Applying trigonometry 23
Further elevation and depression
When looking up at an object, the angle is an angle of elevation. When
looking down at an object, you have an angle of depression.
horizontal at eye level of the man
angle of depression
angle of elevation
horizontal at eye level of the dog
line of sight
Angles of elevation and depression are always measured between the line
of sight and the horizontal.
Sometimes you are not given a labelled diagram to assist you. So you
need to visualise the situation yourself and draw a diagram to help you
get the picture clear. Then you can see what triangle to use.
So you have three main steps in solving trigonometry questions:
• draw a diagram to represent the situation
• locate a right-angled triangle in the diagram
• use trigonometry to calculate the side or angle needed.
24 MS5.1.2 Trigonometry
Follow through the steps in this example. Do your own working in the
margin if you wish.
The angle of elevation of the top of a flagpole from an observer
is 39° when measured 12 m away from the flagpole. The
distance from the ground to the observer’s eyes is 1.8 m.
Calculate the total height of the flagpole.
Solution
12 m
39°
A
CB
E D
You need to calculate the total height of the flag above the
ground (that is, AD).
To find AC, you use the tangent ratio in ∆ABC. To find the
actual height of the flagpole, you add CD to AC.
CD = BE (the height of the observer) = 1.8 m
∴ AD = AC + BE
In ∆ABC, tan B = opposite
adjacent
tan 39° = AC
12AC = 12 × tan 39°
= 9.7 m (correct to 1 decimal place)
∴ AD = 9.7 + 1.8 = 11.5 m
Part 3 Applying trigonometry 25
The actual height of the flagpole is 11.5 m (correct to one
decimal place).
Sometimes it may be necessary to add, or subtract, values from the ones
you calculate using trigonometry to arrive at the answer.
Even when a diagram is provided, it may not include all the information.
Feel free to write on the diagram and include other information that may
help you answer the question.
Finally, look over your answer and ask yourself whether it looks
reasonable for the problem.
In the next activity you will be provided with an incomplete diagram.
Mark on it necessary values so you can answer the questions.
Activity – Further elevation and depression
Try these.
1 From the top of a building 60 m high the angle of depression of a
parked car is found to be 28°. Complete the diagram and calculate
the distance the car is parked from the building.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
26 MS5.1.2 Trigonometry
2 Complete the diagram and use it to calculate the angle of elevation
(to the nearest degree) of the top of a wall 15 m high from a point on
the ground 5 m from the bottom of the wall.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
You have been practising further examples of elevation and depression.
Now check that you can solve these kinds of problems by yourself.
Go to the exercises section and complete Exercise 3.4 – Further elevation
and depression.
Part 3 Applying trigonometry 27
Reviewing trigonometry
There is an enormous number of applications for trigonometry. Of
particular interest is the technique of triangulation that is used in
astronomy to measure the distance to nearby stars, in geography to
measure distances between landmarks, and in satellite navigation
systems.
But regardless of where it is used, the basic definitions of sine, cosine
and tangent are the same.
SOH CAH TOA
sin opphyp cos adj
hyp tan oppadj op
pos
ite
adjacent
hypotenuse
In this session you will practice the ideas you learned in using
trigonometry in right-angled triangles.
The following exercises will help you to consolidate trigonometry
applications.
Go to the exercises section and complete Exercise 3.5 – Reviewing
trigonometry.
28 MS5.1.2 Trigonometry
Part 3 Applying trigonometry 29
Suggested answers – Part 3
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1 a 0.485 b 0.176 c 0.602 d 0.725
2 a 34° b 85° c 81° d 45°
3 cos 36° =x
126x =126 × cos 36°
=101.9 m
4 tan 38° =e
7.6e = 7.6 × tan 38°
= 5.94 cm
5 sin x° =5
6x = 56°
6 tan h =9.5
15.3h = 32°
Activity – Choosing the ratio
1 a tan =opposite
adjacent
tan 50° = x
80x = 80 × tan 50°
=95.3 mm
30 MS5.1.2 Trigonometry
b cos =adjacent
hypotenuse
cos 60° = x
100x = 100 × cos 60°
= 50 m
2 a sin α = 3
8 ∴ α = 22°
b cos α = 9
41 ∴ α = 77°
c tan α = 5
7You can use trig ratios to calculate β.
∴ α = 36° Alternatively, β �= 90°– 36°
=54°
Activity – General problems
1 cos =adjacent
hypotenuse
cos 75° = 1.3
y
y = 1.3
cos 75°= 5.02
The ladder is 5 m long.
2 tan =opposite
adjacent
tan θ = 40
500θ = 85.41°
= 85°
(correct to the nearest degree)
3 The gradient, m = 2 (so tanθ = 2 ) and the angle the line makes is
63° .
Part 3 Applying trigonometry 31
Activity – Elevation and depression
1 tan 15° = 20
SF
SF = 20
tan 15°= 74.6 m
(correct to 1 dec. pl.)
2 Let the angle be θ .
tan θ = 10
16θ = 32°
(to the nearest degree)
Activity – Further elevation and depression
1 tan 28° = 60
d
d = 60
tan 28°= 112.8 m
(correct to 1 dec. pl.)
The car is parked 112.8 m
from the building.
28°
d60
m
28°
2 Let the angle be θ .
tan θ = 15
5θ = 72°
(to the nearest degree)
The angle of elevation is 72° .5 m
15 m
θ
32 MS5.1.2 Trigonometry
Part 3 Applying trigonometry 33
Exercises – Part 3
Exercises 3.1 to 3.5 Name ___________________________
Teacher ___________________________
Exercise 3.1 – Choosing the ratio
1 Choose a suitable trigonometric ratio and then calculate x (measured
in centimetres), correct to two decimal places.
a
40030°
x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
x
40°75
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
34 MS5.1.2 Trigonometry
c
x
25°
8
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
d
A C
B
80
48°x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
e
28
59°
x
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 3 Applying trigonometry 35
2 Calculate the value of α (alpha) in degrees in the following triangles.
a
7
α10
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
b
2
α6
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
c
36
α
9
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
36 MS5.1.2 Trigonometry
Exercise 3.2 – General problems
1 A ladder reaches 2.5 m up a vertical wall, and has its foot on level
ground 1 m from the base of the wall. Find the angle the ladder
makes with the ground.
θ
2.5
m
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
2 A 30° – 60° – 90° set square has the side opposite the 60° angle
12 cm long. Find the length of the longest side. (Give your answer
correct to the nearest millimetre. Hint: label the longest side with a
pronumeral.)
60°
30°
12 c
m
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Part 3 Applying trigonometry 37
3 The diagonal BD of a rectangle ABCD makes an angle of 27° with
the side AB, which is 7.0 cm long. Calculate the length of the other
side, correct to two significant figures.
A B
D C
7.0 cm27°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
4 To calculate the width of a river, a man stands at a point A directly
opposite a tree T on the edge of the opposite bank. He walks 120 m
along his bank to another point B and measures ∠ABT = 27°.
Calculate the width of the river. (The banks are straight and parallel
along this section of the river.)
120 m
A
T
B27°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
38 MS5.1.2 Trigonometry
5 O is the centre of the circle. OD = 3.28 cm, and ∠AOD = 49°.
Calculate the radius of the circle.
O
A D B
r 49°
3.28
cm
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
6 The pitch of a roof is a measure of its steepness. It is often
expressed as the ratio of the rise to the run.
rise
runθ
Drawn to scale
Use a ruler to measure the rise and run on this scale diagram. Use
this information to calculate the pitch angle, θ .
__________________________________________________________
Part 3 Applying trigonometry 39
Exercise 3.3 – Elevation and depression
1 From a boat out at sea, Chandra measured the angle of elevation of
the top of a cliff 65 m high to be 27°. How far is the boat from the
foot of the cliff (to the nearest metre)?65
m
27°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
2 Kimberley found the angle of depression of the base of a statue in a
nearby park to be 65° from a 35 m high window. How far from the
building is the statue? (Answer correct to the nearest metre.)
35 m
65°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
40 MS5.1.2 Trigonometry
3 The altitude of the sun is the angle of elevation of the sun. It is the
angle between the horizontal and the sun’s rays. A building casts a
shadow 40 m long when the altitude of the Sun is 50°. Find the
height of the building, correct to the nearest metre.
40 m50°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
4 Calculate the angle of depression of Mrs Lee’s cottage which is
1.6 km away down a slope from the top of a hill 225 m higher.
1. 6 km
225
m
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Part 3 Applying trigonometry 41
5 Find the elevation of the sun when Olav, who is 1.9 m tall, casts a
shadow 2.6 m long. (Answer to the nearest degree.)
α
2.6 m
1.9
m
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
6 (Harder) You will need to draw a diagram for this one. The angle
of elevation of an aircraft flying at 800 metres above ground level is
64° from a point on the ground. How far is the aircraft from this
point?
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
42 MS5.1.2 Trigonometry
Exercise 3.4 – Further elevation and depression
In each of these only a partial diagram is provided. Label the diagram
appropriately to help you answer the question.
1 From the top of a vertical cliff 40 m above sea level the angle of
depression of a boat is measured to be 18°. Find the distance from
the boat to the bottom of the cliff.
40 m
18°
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
2 An upright stick casts a shadow of length 2 m on level ground. If the
stick is 1 m long, find the angle of elevation of the sun at this time.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Part 3 Applying trigonometry 43
3 From a plane flying at a height of 4 km above a town A, the angle of
depression of a town B is found to be 20°. Find the distance
between the two towns, correct to one tenth of a kilometre.
Town A Town B
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
4 A tent pole is 2.2 metres high. Calculate the angle of elevation of
the top of the pole given the top of the pole is secured to the ground
with a rope 3.5 m long.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
44 MS5.1.2 Trigonometry
5 A boat is at anchor where the top of the 25 m anchor chain is 16 m
above the seabed. Calculate the angle of depression of the anchor
chain.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
6 The Eiffel Tower was built for the Universal Exhibition in
celebration of the French Revolution and opened in 1889.
Write a trigonometry question
you could answer using the
information on this diagram.
(You do not need to answer the
question.)
324 m
200 m
θ
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Part 3 Applying trigonometry 45
Exercise 3.5 – Reviewing trigonometry
1 Calculate the lengths or angles marked.
a
4 cm
x30°
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b
35°
d cm
50 cm
F
D E
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cg
14.6 9.5
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46 MS5.1.2 Trigonometry
d
12 mm
A
B C
7 m
m
α°
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e
5.7 mm
38°
j
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2 A boy is flying a kite on 45 metres of line which makes an angle of
44° with the horizontal. How much higher is the kite than the boy’s
hand? (Answer to the nearest metre.)
45 m
44°
h
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Part 3 Applying trigonometry 47
3 A fishing boat is held at anchor in the water by 23 metres of chain.
The depth of the water is 18 metres. Find the angle the chain makes
with the sea floor.
1823
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4 Calculate the angle the line makes with the x-axis.
–1–2 1 2 30
–1
1
2
3 y
x
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48 MS5.1.2 Trigonometry
5 One end of a 10-metre rope is tied to the bow of a yacht and the
other end to a point on the edge of a jetty. The rope is taut (stretched
tightly) and makes an angle of 21° with the horizontal. How far out
is the bow from the wharf?
21°10 m
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6 A 14-metre fire engine ladder has its foot 4 m away from the side of
a building. Calculate the angle the ladder makes with the wall.
4 m
14 m
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Part 3 Applying trigonometry 49
7 A fish notices a bug on the
surface of the water 55 cm in
front of it, and 65 cm above it.
65 cm
55 cm
a Calculate the angle of elevation of the bug from the fish.
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b How far away is the bug from the fish?
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